For each production function shown below, a)prove algebraically whether the function satisfies increasing, decreasing, or constant returns to scale and b) describe the organization of an industry represented by such a function (for part b, I am expecting you to comment on both the typical firm size as well as the general number of firms expected in the industry) i. ??(??,??) = (4/3)L^.5 + 16KL^.25 ii. ??(??,??) = 2??^2 + L iii. ??(??,L) = K^.33 + 3sqrt KL + 2/3L Solution As the equations are not stated properly few assumptions have been made in the equations. (a) First Function: Q = 4/3L0.5+16(KL)0.25 We will increase both K and L by m and create a new production function Q. Then we will compare Q to Q. Q = 4/3(mL)0.5+16(m2KL)0.25 = (m0.5)4/3L0.5+(m0.5)16(KL)0.25 = (m0.5) (Q) Increasing the factors of production by m, output incrreases by half of m. Hence decreasing return to scale exists. Second Function: Q = 2k2+L Q-L=2k2 m(Q-L) = 2mK2.............eq1 Again increasing both K and L by m and create a new production function Q. Q = 2(km)2+Lm = m(2mk2+L) = m(mQ-mL-L) (from eq1) = m2(Q-L-L/m) Hence increasing return. Third Function : Q = K + 3sqrt KL + 2/3L increasing both K and L by m and create a new production function Q. Q = mK + 3msqrt KL + m2/3L = m(K + 3sqrt KL + 2/3L) = mQ Hence constant return to scale exists. (b) Lets take a public accounting industry. Very largest accounting firms represnts either decreasing or constant return to scale, smaller frims face increasing return to scale..